LimitsOfArbitrage_LATEST.pdf

Who Limits Arbitrage?

∗

Nathan Foley-Fisher Borghan Narajabad Stéphane Verani †

June 2020

Abstract

The cost of selling a security short is an equilibrium outcome of the demand for short positions and the willingness of long-term institutional investors to lend their securities to short sellers. We introduce an imperfectly competitive securities lending market into a model of securities traders with private information. Long-term institutional investors with greater liquidity risk tolerance are more willing to lend their securities, lowering the cost of taking short positions, which increases price informativeness in the spot trading market. We find empirical support for these predictions in using the universe of corporate bond trades and life insurers’ corporate bond loans.

JEL Codes: G11, G22, G23

Keywords: limits of arbitrage, securities lending, life insurers, corporate bonds, market microstructure, strategic trading, liquidity

∗

All authors are in the Research and Statistics Division of the Federal Reserve Board. For providing valuable comments, we would like to thank, without implication, Celso Brunetti, Stefan Gissler, Andrew Metrick, Coco Ramirez, and participants in presentations at the Federal Reserve System Committee on Financial Institutions, Regulation, and Markets 2019, Nova Business School, University of New South Wales, and the Federal Reserve Board. The views in this paper are solely the authors’ and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

†

Introduction

Limits of arbitrage prevent securities prices from immediately and fully reflecting changes in

their fundamental value. Past research has focused on how costly arbitrage prevents traders

from eliminating mispricing, while more recent studies have emphasized the role of financial

institutions and agency frictions (Gromb & Vayanos 2010). Nevertheless, the literature remains

focused on the decisions of direct security spot market participants. This literature overlooks

the decisions of institutions and agents outside of the spot market that can have a significant

role in determining the limits of arbitrage.

We propose a theory of security prices based on a rational expectations strategic equilibrium

framework with simultaneous market clearing in a secondary spot and a securities lending

market. We build on a simplified version of Kyle (1989) and expand the framework to incorporate

short sale restrictions and an imperfectly competitive securities lending market. Traders wishing

to establish a short position must borrow the security from securities lenders by posting cash as

collateral. The key innovation in the model is to endogenize the short-selling cost by considering

the problem of a strategic securities lender. The cost of borrowing the security is an equilibrium

outcome, as potential securities lenders take into account the price effects of their own supply

of the security to short sellers.

The main result is that the informativeness of the security spot price depends on the securities

lenders’ cash collateral reinvestment decisions. In the model, the security spot and securities

lending market are connected by the slope of the securities lenders’ supply curve. Traders in the

spot market wish to take opposite positions when they disagree about the security’s liquidation

value. Short-sellers confront the securities lenders’ supply schedule when implementing their

short position. When the supply of securities from securities lenders is relatively flat,

short-sellers can borrow the security without significantly increasing the short-selling cost.

cash collateral reinvestment risk. The higher is a strategic securities lender’s risk tolerance,

the higher is her willingness to lend the security. However, strategic securities lenders take

into account the effect of their lending decision when deciding to lend to short-sellers, because

supplying more securities decreases the lending fee. A lower security lending fee, in turn,

decreases the risky reinvestment expected return. In equilibrium, the spot market is more

active and the spot price is more informative when the strategic securities lender has a higher

tolerance for reinvestment risk.

Limits of arbitrage arise in the model for two reasons. First, there are well-known limits

because imperfectly competing traders have private information about the security’s value and

internalize the effects of their trading position on the security spot price (Kyle 1989, Vayanos

1999). Second, there are holding costs associated with implementing short positions that prevent

arbitrage trades from fully revealing the true value of the security (Tuckman & Vila 1992,

Dow & Gorton 1994, Duffie 1996, Krishnamurthy 2002, Gromb & Vayanos 2010). We show

that short-selling costs depend on the risk tolerance of imperfectly competing securities lender

that internalize the effect of their lending decisions on their cash collateral reinvestment return.

Although we assume our traders have deep pockets, the limits of arbitrage that arise in the model

are likely to be exacerbated by, for example, leverage constraints that affect the implementation

of long and short trading positions (Brunnermeier & Pedersen 2009, Gorton & Metrick 2012,

Krishnamurthy, Nagel & Orlov 2014, Dow, Han & Sangiorgi 2018).

We test the relationship between securities lenders’ cash collateral reinvestment and the

informativeness of spot market trading using a cross-section of more than 12,000 corporate bonds

held by 109 insurance companies with individual securities lending programs and cash collateral

reinvestment strategies. Our novel dataset combines the universe of corporate bond trading data

from the spot market to the universe of securities lending transaction. We use transaction-level

spot market data from TRACE to obtain a measure of price informativeness for each corporate

transaction from Markit Securities Finance. We then add new annual regulatory data on U.S.

insurance companies that specify the bonds in each insurer’s portfolio that are on loan at the

time of filing as well as security-level data on each insurer’s cash collateral reinvestment portfolio.

Crucially, we observe precisely how each individual life insurers reinvest the cash collateral they

receive in exchange for the bonds that they lend to short-sellers. These new data allow us to

construct a measure of the cash collateral reinvestment risk of every insurance company that

has a securities lending program.

We first show that price informativeness in the spot market is correlated with the cash

collateral reinvestment strategies of life insurers holding that bond. This correlation is consistent

with the key predictions of our model. We also show that bonds with greater trade volume in

the spot market are disproportional held by by life insurers that reinvest their overnight cash

collateral more aggressively in long-term or high yield assets.

We then identify the mechanism of our model by studying the actual decisions of individual

insurers to lend specific corporate bonds. We show that life insurers with more aggressive cash

collateral reinvestment strategies are more likely to lend thesame bond at the same time when

they face higher demand for that bond from short sellers. We identify this effect by exploiting

the rich cross-sectional variation in cash collateral reinvestment strategies of life insurers holding

the same bond and facing the same demand from short sellers at the same time. We show that

securities lenders with more aggressive cash collateral reinvestment strategies relax the limits

of arbitrage by lending to short sellers at lower fees and thus reduce the cost of taking short

positions. Taken together, our results show that life insurers with more aggressive securities

lending programs relax the limits of arbitrage.

In addition to the contribution to the theoretical literature described above, our empirical

results contribute to the literature that identifies the securities lending market as a potential

source of limits to arbitrage. Existing studies have documented regulations (Diether, Lee

structure (Nagel 2005, Hirshleifer, Teoh & Yu 2011, Porras Prado, Saffi & Sturgess 2016,

Foley-Fisher, Gissler & Verani 2019), the costs of short selling (Jones & Lamont 2002, Geczy, Musto &

Reed 2002, Asquith, Au, Covert & Pathak 2013, Cohen, Diether & Malloy 2007, Engelberg, Reed

& Ringgenberg 2018), and the availability of traded options (Figlewski & Webb 1993, Danielsen

& Sorescu 2001) as having an effect on spot market trading through the securities lending

market. Much of this literature identifies exogenous determinants of the limits to arbitrage.

Our key empirical contribution to this literature is to document compelling evidence that the

strategic decisions of securities lenders to reinvest their cash collateral in risky portfolios is an

endogenous determinant of the limits to arbitrage.

The remainder of the paper proceeds as follows. In Section 2, we present the model. Section 1

provides an overview of the market for lending securities. Section 3 describes our data and

provides summary statistics. Sections 4 present our empirical results.

1

The role of securities lenders in corporate bond trading

In this section, we first outline the role of securities lending in corporate bond market trading.

We describe a typical corporate bond trade transaction, including the specific role played by

securities lenders. Then, we outline the typical structure of a securities lending transaction,

together with the motivations of each party to the deal.

1.1 Corporate bond trading

Corporate bonds are traded in a large over-the-counter (OTC) market. In 2015, U.S.

corporations issued almost $1,500 billion of debt, compared with only $256 billion in equity.1

After their initial offering in the primary market, most of this debt is tradable in an OTC spot

market. In 2015, over 25,000 unique corporate bonds were publicly traded, with most of the

trading taking place in investment grade bonds (61 percent). Between [[2011 and 2015, there

1

were 44,082 daily transactions on average that amounted to almost $30 billion in daily volume

traded.]] About two-thirds of these transactions were between a client and a dealer, and the

other third of these transactions were between dealers.2

To buy and sell corporate bonds on the secondary market, intermediaries, such as

broker-dealers, typically facilitate trading by either swiftly finding a matching counterparty for a client

in the market, or by trading itself with the client and maintaining its own inventory of bonds.

Dealers inventories are naturally limited by the supply of individual bonds and the associated

inventory maintenance costs.3 The limitations on dealers’ ability to fulfill client orders create an

opportunity for buy-and-hold institutional investors, such as insurance companies and pension

funds, as natural large repositories of securities to lend their securities. Among institutional

investors in corporate bonds, insurance companies have the largest holdings, giving them a

dominant position as potential bond lenders.4 When a client wants to buy a bond that a dealer

does not hold in its inventory, the dealer may borrow the bond elsewhere and deliver it to the

buyer. The dealer can then wait until it can find another client willing to sell the same bond,

which the dealer can return to the lender.

Corporate bond market participants, such as dealers and hedge funds, have a variety of

reasons for borrowing a particular bond. It is common to borrow a bond to take a short position

or to cover a naked short position (Duffie 1996, Merrick Jr, Naik & Yadav 2005). In addition to

concerns about about future bond yields, bond short-sellers may be concerned with individual

firm performance. For example, in a capital structure arbitrage trade, a firm’s bond is shorted

to hedge a long position in the firm equity (Liu & Mello 2011). Another example is a convertible

bond arbitrage trade, in which firm’s equity is sold short to hedge a long position in a bond

issued by that firm (Agarwal, Fung, Loon & Naik 2011). In this second example, the hedge fund

2

www.finra.org.All these statistics exclude convertible debt transactions.

3 _{Dealer inventory cost may be affected by regulations, such as the Volcker Rule that restricts proprietary} trading by banks (Bao, O’Hara & Zhou 2018, Schultz 2017, Dick-Nielsen & Rossi 2018).

borrows the stock and the dealer might also borrow the convertible bond.

1.2 Securities lending and cash collateral reinvestment

In a prototypical corporate bond loan, full legal and economic ownership of the bond is

transferred from the lender (e.g. insurance company) to the borrower. The ownership is essential

for borrowers (e.g. dealers) to be able to deliver the bond to other counterparties (clients). To

allow the borrower flexibility in the time needed to find another seller of the same bond, the

term of the loan is usually open-ended, but either party is able to terminate the deal at any

time by returning the security/collateral.5

In exchange, the bond borrower gives the bond lender collateral in the form of cash, which

the lender may reinvest according to its own strategy and regulatory limitations.6 Typically, the

loan is marked to market daily and is “overcollateralized,” with borrowers providing, for example,

$102 in cash for every $100 in notional value of a security. The percentage of overcollateralization

is called the “margin,” which serves to insure the securities lender against the cost of replacing

the lent security if the borrower defaults. Lastly, the bond lender pays a percentage of the

reinvestment income to the bond borrower, called the “rebate rate.” This equilibrium price is

negotiated at the outset of the deal and reflects the scarcity of the bond on loan: A hard-to-find

“special” bond may command a low or negative rebate rate.

The ultimate securities owners decide which securities in their portfolios will be made

available to lend and how the cash collateral proceeds of their lending programs will be reinvested

(Keane 2013, Foley-Fisher, Narajabad & Verani 2019).7 Some institutional investors lend only

opportunistically, lending securities that are special and reinvesting the cash collateral in

cash-like debt instrument, such as treasury and agency repo. Others institutional investors may lend

5

Even in the unusual cases of term lending, parties often have the ability to break the contract early by paying a nominal penalty. More than 90 percent of the corporate bond loans in our data sample are open-ended. 6 _{In principle, the contract may allow a borrower to post non-cash collateral against the bond, but this is} uncommon in the U.S. In our data on corporate bond loans, more than 90 percent of transactions are against cash collateral.

7

out a greater portion of their portfolio regardless of the rebate and reinvest the cash collateral

in a portfolio of relatively illiquid assets. How aggressively a securities lender pursue

securities-lending income depends on its risk tolerance. This suggests a non-trivial relationship between

short-selling costs and securities lenders’ cash collateral reinvestment strategy.8 This relationship

is likely to be more relevant for thin markets (Rostek & Weretka 2016), where, for example, life

insurers hold a large fraction of individual corporate bond issue. In the next section, we propose

a new theory to formalize the link between an institutional investor securities lending strategy

and spot market trading.

2

A theory of security prices with endogenous short-selling costs

We propose a theory of security prices based on a rational expectations strategic equilibrium

framework with double auctions in the spot market and the securities lending market. We start

from a simplified version of the model introduced in Kyle (1989) that we expand to incorporate

short-sale restrictions and an imperfectly competitive securities lending market. Traders wishing

to establish a short position must borrow the security from securities lenders and post cash as

collateral. From the perspective of the short-seller, the securities lending transaction corresponds

to a reverse repurchase agreement.9 The cost of borrowing the security affects the short-selling

cost and is an equilibrium outcome, as potential securities lenders’ take into account the price

effects of their own supply of the security to short sellers. The endogenous short-selling cost

affects traders’ short positions, and, therefore, the informativeness of the security price in the

spot market. Limits of arbitrage arise because traders and securities lenders rationally internalize

the price effects of their position in the spot market and in the securities lending market.

8

Focusing on the 2008-09 financial crisis, Foley-Fisher, Gissler & Verani (2019) shows corporate bonds liquidity was adversely affected by the shutdown of AIG’s securities lending program after the company experienced a run on its securities lending program.

9

2.1 Market structure

There is a single risky security traded in a single period. Let ν ∼N(0, τ_ν−1) be the normalized

random liquidation value of the security that is realized after the one-period trading session.

2.1.1 Spot market participants

Two types of traders participate in the spot market. All traders have a non-stochastic

endowment, which is normalized to zero. A single privately informed trader trades strategically

against the residual demand from a large number of small, privately informed non-strategic

traders, whose combined position is assumed to be exogenous.

The first type of trader is a single strategic informed trader who receives a private signal

s=ν+ε about the value of the security, whereε∼N(0, τ_ε−1) is random noise. The informed

trader has an exponential utility function with risk-aversion parameterρi given by

−exp (−ρi·(ν−pˆ)·x) , (1)

wherex ∈Rdenotes her trading position and pˆdenotes the per unit cost of taking a position.

These quantities and prices are determined in equilibrium.

The second type of trader is a unit measure continuum of atomistic non-strategic chartists.

These traders’ combined position in the trading session is random and exogenously given by

(z−pˆ)·κ , (2)

where z = ν + ζ and ζ ∼ N(0, τ_ζ−1). We use the label “chartists” for the second type of

traders because they implement a position that is proportional to the difference between their

benchmark, z, and the cost of the position, pˆ. The parameter κ is the slope of the residual

demand from the chartists’ combined position.10

The unit cost of trading, pˆ, depends on the direction of the position. When taking a long

position, traders pay the spot price p to acquire the security. When taking a short position,

traders need to borrow the security from a securities lender by posting cash collateral. We

follow the institutional setup of the rebate, described in Section 1, when modeling the cost of

borrowing the security, which we denote byr. Therebate is the percentage of the cash collateral

reinvestment income that the securities lender pay to the securities borrower. The rebater is a

non-positive amount in our model because we normalize the risk-free interest rate to zero. The

rebate is more negative when the security is in high demand from short-sellers. The net per

unit short-selling cost is therefore p+r. When a trader thinks the security is overvalued and

would like to take a short position, a more negative rebaterdecreases the attractiveness of short

selling, asp+r is lower. In summary, the cost of trading is given by:

ˆ p=

   

  

p if x≥0

p+r if x <0.

2.1.2 Securities lending market participants

Traders that borrow securities to take a short position participate in the securities lending

market together with two types of potential securities lenders. Both types of securities lenders

are endowed with an unlimited and unconstrained stock of the security. A single potential

securities lender has access to a risky cash collateral reinvestment opportunity and strategically

chooses how much of the security to lend taking as given the combined supply of the security

supplied to short-sellers by a large number of small and non-strategic potential securities lenders.

The first type of securities lender is a unit measure continuum of atomistic non-strategic

securities lenders supplying Ln = −`n·r units of the security to short-sellers, where <sub>`</sub>1_n > 0

denotes the slope of the non-strategic securities lenders’ aggregate supply curve. When the

demand from short-sellers is high, the non-strategic security lenders respond to the more negative

rebate r by increasing their supply of the security. If the supply curve steep, `n is low, a high

demand for borrowing the security results in a more negative rebate rate, r, which in turn, as

we discussed above, decreases the attractiveness of short selling.11

The second type of securities lender is a single strategic securities lender who reinvests the

cash collateral in a risky project with returnR∼N 0, τ_R−1. The strategic securities lender has

an exponential utility function with risk-aversion parameterρs given by

−exp (−ρs·(R−r)·Ls) , (3)

whereLs is the quantity of the security that is lent to short-sellers.

2.2 Equilibrium

Any equilibrium requires the spot and securities lending markets to clear simultaneously. An

intimate connection between the spot market and securities lending market arises because traders

wishing to take short positions in the spot market need to borrow the security in the securities

lending market. The strategic securities lender’s supply affects the positions of the spot market

traders through the rebate, which is the short-selling cost. At the same time, the informed

trader’s position can affect the rebate through her demand for borrowing the security.

We follow Kyle (1989) and focus on existence and characteristics of the linear Nash

equilibrium. Specifically, we assume that the strategic informed trader and the strategic

securities lender consider the effects of their decisions on equilibrium prices to be linear. We

then characterize their optimal decision and verify that the market clearing prices are indeed

linear in the decisions of the informed trader and the strategic securities lenders.

11

2.2.1 Securities spot market

We begin by holding the supply of security and the rebate constant. The informed trader chooses

her trading positionx to maximize her expected utility, which solves the following problem:

max

x E[−exp (−ρi·(ν−pˆ)·x)|s, p, r] (4)

s.t. pˆ=       

p x≥0

p+r x <0.

The first order condition for the solution of (4) implies that

x = E(ν|s, p, r)−pˆ

VAR(ν|s, p, r) + ˆp0(x).

(5)

The marginal effect of the informed trader’s position on the cost of the trading position, pˆ0(x),

depends on the security equilibrium spot price and rebate.

The spot market clears when x+ (z−p)·κ= 0, which implies thatp=z+x_κ. We assume

that the rebater is linear in the trading positionx, and that ifx= 0 thenr= 0. We verify that

this is indeed the case in Section 2.2.2 below. The linearity of r and the spot market clearing

condition imply that r =−|x<sub>`</sub>| for a positive `. Therefore, the informed trader’s marginal cost

of taking a long position x≥0and short position x <0is given by

ˆ

p0(x) =        1

κ x≥0

1

κ +

1

` x <0

. (6)

Note that when taking a short position,x <0, the informed trader takes into account the effect

of her position on both the security spot price p and rebater.

It remains to determine E(ν|s, p, r) and VAR(ν|s, p, r) to fully characterize the solution

of the informed trader’s problem (5). Note that the information content of s, p, and r for the

traders’ benchmark,s=ν+εandz=ν+ζ, respectively. Given thatεandζ are independent,

this implies that the trader’s learning—i.e., Bayesian inference—about the liquidation value is

given by:

E(ν|s, p, r) =E(ν|s, z) = τε

τν+τε+τζ

·s+ τζ τν+τε+τζ

·z

= τε τν+τε+τζ

·s+ τζ τν+τε+τζ

·

ˆ p−x·

1 κ + 1 ` , (7)

where the second equality follows from the traders’ inference of about ν from the noisy signal,

and the last equality follows from the spot market clearing condition. Similarly, the precision of

the traders’ inference about the true liquidation value is given by

VAR(ν|s, p, r) =VAR(ν|s, z) = 1

τν +τε+τζ

. (8)

Plugging (6), (7), and (8) into (5) yields the informed trader’s optimal trading position as a

function of the endogenous security spot price and security borrowing cost.

x = τε·s−(τν+τε)·pˆ ρi+ (τν+τε+ 2τζ) 1_κ +1_`

, (9)

where the cost of taking the position, pˆ, depends on the direction of the informed trader’s

position: ˆ p =       

p x≥0

p+x_` x <0

. (10)

Because the informed trader takes into account the effect of her short position on the cost of

borrowing the security, r= x_`, the informed trader’ short position is given by:

x = − (τν+τε)·p−τε·s

ρi+ (τν +τε+ 2τζ)_κ1 + 2 (τν+τε+τζ)1_`

In the model, short-sellers are thesupply side of the spot market. For example, the chartists

take a long position when their benchmark, z, is more optimistic than the informed trader’s

signal, s, and the informed trader takes the opposite position and shorts the security. The

chartists’ position, (z−p)·κ is the downward sloping demand schedule in the spot market and

the informed trader’s short position,−x, is the upward slopping supply schedule. It follows that

the equilibrium security spot price necessarily falls between sandz.

Equation (11) shows that the slope of the supply of securities in the securities lending market,

`, directly affects the slope of the upward sloping supply schedule in the spot market. For

example, the supply schedule of the spot market is relatively flat when the supply of securities

in the securities lending market is also relatively flat. In this case, short sellers can borrow the

security in relatively large proportion without significantly increasing the short-selling cost.

Figure 1 illustrates how the slope of the supply of securities in the securities lending market

affects the equilibrium spot price. In this example, we assume that the informed trader shorts

the security, x < 0 and the realization of ν and is such that ν = = 0.12 We also assume

that κ = 1, which means that the residual chartist demand is unit elastic. Figure 1 plots the

supply schedules in the spot market corresponding to high and low levels of security supply from

securities lenders, `, and demand schedules for high and low levels of the chartists’ benchmark

value about the security, z. Each of the four black dots represent an equilibrium security

spot price and quantity for a given level of ` and z. The equilibrium spot price and quantity

corresponding to a relatively low value of`andzis represented by the point(p1, x1). A steeper

supply curve in the securities lending market yields a greater and lower equilibrium spot price

and quantity, which is represented by (p2, x2). This is because a lower ` means that it is more

expensive for short sellers to borrow the security from securities lenders. Because implementing

the short position is more costly, it is also more revealing of the signal received by the informed

trader. As a result, there is less trade and the spot price is higher. This effect is larger when the

12 _{The other model parameters are set as follows: ρ}

i = 0.1; `n = 1; τν =τ =τζ = τR = 1; ν = 0;= 0;

chartists are more optimistic and their benchmark security valuezis higher, which is represented

by a move from (p0₁, x0₁) to (p0₂, x0₂). In this case the price effect of the short-seller is greater,

which leads to an even less revealing security spot price.

To find the equilibrium quantity and price of security traded in the spot market, we use the

spot market clearing condition in equation (9) to obtain:

x∗ = τε·s−(τν +τε)·z ρi+ 2 (τν+τε+τζ)· _κ1 +1_`

p∗ =        ˆ

p∗ x≥0

ˆ p∗−x∗

` x <0

, (12)

where

ˆ

p∗ = τε

1

κ +

1 `

·s+ ρi+ (τν +τε+ 2τζ) 1_κ+ 1_`

·z ρi+ 2 (τν+τε+τζ)· 1_κ +1_`

.

Equation (12) illustrates further the effect of securities lending on the equilibrium quantity

and price of security traded in the spot market. When there is a lot of disagreement among

traders about the liquidation value of the security—i.e., | s−z | is large—traders want to

take opposite positions. When the supply of securities by securities lender is flat, 1_` is small,

traders can borrow the security without significantly increasing the short-selling cost. In this

case, the spot market is more active, and the spot price is more informative about the security’s

liquidation value. In the limit `→ ∞, thus 1<sub>`</sub> → 0, our model collapses to a simple version of

Kyle (1989). In this special case, arbitrage is limited because the disagreement among traders

means that trading in the spot market does not fully reveal the true liquidation value. We turn

2.2.2 Securities lending market

Proposition 2.1 explains how the supply of securities by securities lenders affect spot market

trading.

Proposition 2.1 The expected trading volumeE(|x∗|) and the informativeness of the security

price _dνd_E(p∗|ν) in the spot market are decreasing in the slope of the securities lending supply, 1_`.

Proof The proof follows from substitutings=ν+εandz=ν+ζ in equation (12) and noting

that when x >0

d dνE(p

∗_|

ν) = 1− _ρi τν

1

κ+

1

`

+ 2 (τν +τε+τζ)

, (13)

and whenx <0

d dνE(p

∗_|

ν) = 1− τν

1− 1

1 κ+ 1 ` ρi 1 κ+ 1 `

+ 2 (τν+τε+τζ)

. (14)

The spot and securities lending market are connected by the slope of the securities lenders’

supply curve, 1_`. To solve for an equilibrium, we need to characterize the solution for the strategic

lender’s problem, because the total supply of security to short sellers, −`·r, is the sum of the

security supplied by the non-strategic lenders,Ln=−`n·r, and by the strategic lender,Ls. We

also need to verify that the optimal strategic securities supply function is linear, as previously

assumed.

The strategic lender chooses how much security to lend to short sellers to maximize his

expected profit given in problem (3). In doing so, the strategic lender takes into account the

effect of his lending decision on the rebate r. The first-order-condition for problem (3) is

which implies that

Ls = E

(R|r)−r(Ls) ρs·VAR(R|r) +r0(Ls)

= −_ρ r s

τR +

1 `

. (15)

The second equality follows from the assumption that the strategic securities lender’s

reinvestment project and the short interest are independent.

We check that the securities lending market clears to verify that the strategic lender’s linear

supply—equation (15)— is indeed optimal. Market clearing requires that,

−`·r = −`n·r+Ls

= −`n·r− r ρs

τR +

1 `

,

which implies that the slope of the securities lenders’ supply curve is

` = `n 2 ·

1 +

r

1 + 4τR ρs·`n

. (16)

Proposition 2.2 below follows immediately.

Proposition 2.2 The slope of of the securities lenders’ supply curve, 1_`, is increasing in the

slope of the non-strategic lender’s supply curve, _`1

n, and is decreasing in the strategic lender’s

risk tolerance,1/ρs.

Proof The proof follows from (16).

Figure 2 and 3 plots the equilibrium spot quantity and price as a function of the the strategic

securities lender’s risk tolerance,1/ρs, respectively. Recall that equation (16) implies that that

plotted in Figure 1. The quantity traded in the spot market is increasing in the securities lenders’

risk tolerance, while the security spot price is decreasing in the securities lenders’ risk tolerance.

The strategic securities lender with a higher risk tolerance for reinvestment risk—i.e., a

lowerρs— has a higher willingness to lend the security. In this case, an increase in short interest

does not result in a significant increase in short-selling cost. Conversely, arbitrage could be

severely limited if the strategic securities lender’s risk aversion is very high. Corollary 2.3 below

summarizes this property.

Corollary 2.3 The expected trading volume _E(|x∗ |) and the informativeness of the security

price, _dνdE(p∗|ν), are decreasing in the strategic securities lender coefficient of risk-aversion, ρs.

Proof The proof follows from propositions (2.1) and (2.2).

Figure 4 plots the informativeness of the security spot price as a function of the the strategic

securities lender’s risk tolerance, 1/ρs. When the strategic lender risk tolerance is very high—

i.e., 1/ρs → ∞— the slope of the securities lenders’ supply curve becomes flat—i.e., 1/`→ 0.

Proposition (2.1) shows that the security spot price is more informative about the security

liquidation value for lower value of1/`. As we discussed above, our model collapses to a simple

version of Kyle (1989) in the limit where 1/ρs → ∞ and 1/` → 0. Therefore, the distance

between the blue solid line to the 100 percent level in Figure 4 when1/ρs→ ∞ is precisely the

effects of the Kyle (1989).

Our theory suggests a novel link between securities lenders’ cash collateral reinvestment and

the informativeness of spot market trading. As in Kyle (1989), natural limits of arbitrage arise

because imperfectly competing traders receive different signals about the security’s liquidation

value and internalize the effects of their trading position on the spot price. When traders need

to cover their short position by borrowing the security against cash collateral, the short-selling

cost affects their position. The short-selling cost depends on imperfectly competing lenders that

more actively traded and its price is more informative when potential securities lenders have a

higher tolerance for reinvestment risk.

The model predicts that the informativeness of the security price in the spot market depends

on the securities lender reinvestment decision, which is depicted in Figure 4. This testable

implication of the model highlights the potentially important role of securities lending market

for liquidity in the spot market. It is clear that testing the implication of the model requires

price and quantity data for securities spot and lending market, as well as some information

about securities lenders’ risk tolerance. The remainder of the paper discuss how we constructed

this data for the universe of U.S. corporate bonds and implemented our tests.

3

Data sources and key variables

We combine several data sources to obtain the dataset we use in our analysis. This section

first lays out how we combine data from the corporate bond spot trading market, the securities

lending market, and insurer’s Statutory filings. We then discuss the key variables that we use

in our analysis and present some summary statistics.

Our data on corporate bond spot trading come from TRACE, created by the Financial

Regulatory Authority (FINRA). Under regulations introduced in 2002 by FINRA, dealers are

required to file detailed reports of their transactions, including trade time, quantity, price, and

counterparty. We follow standard procedures for cleaning these data, including deleting all small

noise-generating trades below $10,000 and removing duplicate transactions.13 We aggregate the

data to a daily frequency by constructing transaction-weighted average prices. Using these daily

data, we calculate the median daily price for each corporate bond over a 28-day period around

the end of each year.

We merge the TRACE data with Mergent’s Fixed Income Securities Database (FISD) by

CUSIP identifier to obtain corporate bond characteristics, including offering amount, offering

13

yield, amount outstanding, credit rating, and a range of indicators on the type of each bond. We

exclude from our sample all bonds that are convertible, putable, privately-placed, asset-backed,

or sold as part of a unit deal. We account for reissuance and early retirement when computing

the amount outstanding over time. We define rating changes using the date of the first action

by a rating agency (Ellul, Jotikasthira & Lundblad 2011).

We match the TRACE data with securities lending market data from Markit Securities

Finance (MSF) dataset using CUSIP identifiers.14 These data include both equity and fixed

income loans and cover about 85 percent of the global market and more than 90 percent of

the U.S. securities lending market. Securities lenders report information about each loan they

have outstanding on a given day, including the security on loan, the value of the loan, duration,

lending fee, rebate rate, and the type of collateral posted. In addition, securities lenders report

on each day the total value of every security that they have available to lend. We first aggregate

these transaction-level data to a daily frequency by calculating for each corporate bond the

total amount available for borrowing, the total amount on loan, and the transaction-weighted

average rebate. Using these daily measures, we compute the median daily values for the amount

available, the amount on loan, and the rebate for each corporate bond over a 28-day period

around the end of each year.

Lastly, we combine our TRACE-FISD-MSF merged dataset with information about

insurance companies’ corporate bond holdings, securities lending decisions, and cash collateral

reinvestment portfolios from the NAIC Annual Statutory Filings. This information is reported

as of the end of each year over the period from 2011 through to 2015. Within these filings,

Schedule D reports each life insurer’s holdings of individual fixed income securities, together with

cross-sectional information about each security, including the CUSIP identifier and whether the

bond was on loan as part of the insurer’s securities lending program or subject to a repurchase

14

agreement.15 We drew information about the total size and performance of the insurer’s

investment portfolio from the summary balance sheet. We focus on all insurance companies

that had a securities lending program at any point during our sample period.

The NAIC Quarterly and Annual Statutory Filings also contain Schedule DL, a relatively new

report of individual investments made by insurers using cash collateral received from securities

lending, both on-balance sheet and off-balance sheet.16 _{The Schedule DL was introduced in}

2010 as one of many changes to the reporting and statutory accounting of securities lending

transactions adopted as a response to the 2008-09 financial crisis.17 Figure 5 shows an extract

from one life insurer’s filing in 2012 showing a sample of the individual investments made using

cash collateral received in exchange for lending securities. In general, these new data allow us

to better track the securities lending transactions entered into by an insurer and to observe

detailed information about the life insurers’ use of the collateral received. For example, from

2010, if the collateral received from securities lending could “be sold or pledged by custom or

contract by the reporting entity or its agent,” then the reinvested collateral should be recorded

on the balance sheet.18 We hand-coded data about the maturity of the collateral received in

the securities lending transactions from the regulatory Note 5(e) to the Financial Statements.

Figure 6 shows the relevant notes for the same 2012 sample regulatory filing. Because we rely

on the detailed information collected as part of the new reporting requirements, our sample by

necessity begins in 2011.

15 _{The amount of each bond on loan is not reported. The statutory filings only indicate whether or not the} bond is on loan as of each yearend.

16_{We also use Schedule DL in Foley-Fisher, Narajabad & Verani (2019) in which we study the supply side of} the securities lending market.

17 _{The new guidelines stem from a review of the securities lending practices at AIG that contributed to its} collapse during the 2007-09 financial crisis. In particular, the guidelines specify that borrowers should post cash in the amount of at least 102 percent of domestic securities borrowed (and at least 105 percent if the securities are foreign), that individual loans should not be more than 5 percent of admitted assets, that cash reinvestment should be “prudent,” and that all cash reinvestment securities (on- and off-balance sheet) are reported in the NAIC Quarterly and Annual Statutory Filing Schedule DL. In addition, each asset financed with cash collateral recorded in the NAIC Quarterly and Annual Statutory Filing Schedule D attracts a risk-based capital charge consistent with its NAIC designation code.

http://www.dfs.ny.gov/insurance/circltr/2010/cl2010_16.htm http://www.naic.org/capital_markets_archive/110708.htm

18

Our final merged dataset covers five yearends (2011-2015) with information on 12,411 unique

bonds. We identify 109 insurance companies with securities lending programs. In 2015, this set

of insurers held more than $2.5 trillion in general account assets, or about 43 percent of the

industry total.

3.1 Key variables and summary statistics

The key variables for our study are the price informativeness of each bond in its spot market

and the riskiness of the cash collateral reinvestment strategies of insurance companies that

have securities lending programs. We now describe how we measure these two variables in our

data. Summary statistics for all the variables in our dataset are reported in Table 1. The table is

divided in two, with summary statistics for bond-year data aggregated over insurance companies

reported in the upper panel and summary statistics for bond-insurer-year data reported in the

lower panel.

The concept of bond price informativeness in our model is closely related to an empirical

measure of the bid-ask spread on spot market transactions. Intuitively, when traders with

different information sets can more easily trade in the spot market, the bid-ask spread will be

tighter because movements in the price will be more informative about changes in the underlying

value of the security.19 By contrast, when traders are unable to trade in the spot market, for

example because a securities lender does not make its asset available for short positions, then

movements in the equilibrium market price will not be as informative about changes in the

underlying value of the security and the bid-ask spread will be wider.

We exploit the inverse relationship between the bid-ask spread and price informativeness to

use the label of price informativeness for the negative of the bid-ask spread in the remainder

of our analysis. We take the negative of the spread to make the interpretation of the sign

of the coefficients easier: The transformed variable is increasing in price informativeness. To

19

construct our empirical measure of price informativeness as the negative of the bid-ask spread,

we first calculate for each bond on each day the volume-weighted average buy and sell prices

between dealers and their clients. We then compute price informativeness as the negative of the

average realized bid-ask spread, which is the difference between the average daily price at which

a dealer sells a bond to a client and the average daily price at which a dealer buys the same

bond from a client. We compute the median daily price informativeness for each bond over the

28-day period around the end of each year. Line 6 of Table 1 reports summary statistics for price

informativeness, indicating that in our sample of corporate bonds the average is -0.64 percentage

points with a standard deviation of 0.81 percentage points.

Our proxy for the riskiness of an insurer’s cash collateral reinvestment strategy is based

on the residual maturity that is reported for all types of securities in the regulatory filings.

Specifically, we first calculate the fraction of assets in an insurer’s cash collateral reinvestment

portfolio that have a residual maturity of more than one year. We then subtract the fraction of

cash collateral that is received by the insurer for a duration of more than one year. The one-year

threshold in our calculation is not crucial for the results in the paper. Rather, we choose it so

that our variable represents the investment by life insurers in assets that MMFs cannot purchase

for regulatory reasons.20 It follows that these assets are likely to offer a higher return than cash

instruments.

For each bond i at the end of yeart, we construct a bond-level index for the reinvestment

risk of insurance companies that have securities lending programs. Letmjtbe the fraction of life

insurerj’s cash collateral reinvestment portfolio at the end of yeartthat has a residual maturity

of more than one year. Let hijt be the amount of bond i held by life insurer j at the end of

year t. We construct an insurer-only reinvestment risk index as: P

jhijtmjt/Pjhijt. We also

construct a full-lender index assuming that all other securities lenders of a bond do not have

20

any reinvestment risk (cash collateral maturity transformation).21 LetHjt be the total amount

of bondj that is available for securities lending transactions at the end of yeart. We construct

the all-lender reinvestment risk index as: P

jhijtmjt/Hjt. Summary statistics for these indexes

are reported in the first two lines of Table 1. Across all bonds and years, the mean values of

the two reinvestment risk indexes are 0.18 and 0.13, with standard deviations of 0.16 and 0.17,

respectively.

4

Empirical analysis and results

In this section, we test the empirical predictions of the model of bond trading and lending

that was presented in Section 2. We first study data on individual bonds aggregated over life

insurance companies at the end of each year, as described in the previous section. We show that

price informativeness of an individual bond is correlated with the cash collateral reinvestment

strategies of the life insurers holding that bond. We then study data on insurance companies’

decisions to lend individual bonds in their portfolio. We show that life insurers that reinvest

their cash collateral more aggressively are more willing to lend bonds when there is high demand

from short sellers.

4.1 Spot market price informativeness is correlated with securities lending

Consistent with the results in Section 2, Table 2 shows a robust correlation between the

price informativeness of an individual bond (Price informativeness_it) and the reinvestment

strategy of life insurers that hold the bond. The positive coefficient suggests that greater price

informativeness is associated with greater maturity transformation by life insurers. Column 1

reports the simple correlation using the insurer-only reinvestment risk index. Column 2 reports

the correlation using the full-lender reinvestment risk index and controls for the share of the bond

21

held by life insurers that have securities lending programs (Insurers share_it = P

jhijt/Hjt).

Column 3 adds control variables for the specific market for bond i. Column 4 adds controls for

the type of bond. In all cases, we include year fixed effects to absorb aggregate market conditions

and to focus our analysis on the cross-section of bonds.22

Table 3 repeats the regression analysis of the previous table using the volume of trade in the

spot market as the dependent variable. Reinforcing our previous results, we find a robust positive

relationship between volume (Volumeit) and the reinvestment strategy of life insurers that hold

the bond. Column 1 reports the simple correlation using the insurer-only reinvestment risk index.

Column 2 reports the correlation using the full-lender index and controls for the share of the

bond held by life insurers that have securities lending programs (Insurers shareit=P_jhijt/Hjt).

Column 3 adds control variables for the specific market for bond i. Column 4 adds controls

for the type of bond. We continue to include year fixed effects as in the previous regression

specifications.

Our results so far show that life insurers’ cash collateral reinvestment strategies have an effect

on price informativeness and trading in the secondary spot market for corporate bonds. These

bond-level results are consistent with our model’s predictions. We now test the theoretical

mechanism behind our empirical findings by studying life insurers’ decisions to lend specific

corporate bonds in their portfolio. We use our microdata to study the decision of life insurerj

to lend bondiat timet.

4.2 Limits of arbitrage arising from cash collateral reinvestment strategies

In this section, we show that life insurers with more aggressive cash collateral reinvestment

strategies are more likely to lend the same bond at the same time when they face higher demand

from short sellers. We can identify this effect by exploiting the rich cross-sectional variation in

22

cash collateral reinvestment strategies at life insurers holding the same bond at the same time.

Table 4 summarizes our main result, consistent with the predictions of the model, that

securities lenders that reinvest their cash collateral aggressively are more likely to lend bonds

that have ahigher rebate. The dependent variable is a dummy variable for whether insurerj is

lending bondiat the end of yeart(Loanijt).23 Our main explanatory variables are the fraction

of insurer j’s cash collateral reinvestment portfolio that has a remaining maturity of more than

one year (Reinvestment riskjt) and the weighted average rebate on bondi(Rebateit). Column 1

of the table reports the simple correlation between the variables. The interaction between a

life insurer’s cash collateral reinvestment strategy and a bond’s rebate is positively associated

with the insurer’s decision to lend that particular bond. The association is statistically and

economically significant. Bonds with a higher rebate are less profitable for securities lenders

because the lender must return a greater amount of the reinvestment return to the securities

borrower.

This empirical result survives a battery of controls for the type of bond being lent and its

market conditions. Column 2 reports the results from adding control variables for the percentage

of the bond held by insurers, the concentration of holdings among insurers, and the total amount

on loan relative to the amount that could be lent. In addition, we control for the remaining

maturity of the bond, the offering yield, the offering amount, the amount outstanding, the credit

rating, and the bond issuer (CUSIP6). Going further, Column 3 replaces the bond and market

control variables with bond-time fixed effects that absorb the control variables used in Column 2

as well as unobserved bond-time specific heterogeneity related to the decision to lend bond iat

timet. Column 4 replaces the robust standard errors with errors two-way clustered by insurer

and year.

Lastly, the model predicts that information from the spot market only enters a securities

lender’s decision to lend through its effect on the rebate. That is, the rebate is a sufficient statistic

for securities lenders; they do not need to know about disagreement in the spot market. Column 5

of Table 4 tests this prediction by adding a new interaction term. In a first stage, we regress

individual corporate bond price informativeness on the rebate and take the residuals from the

regression (PI Residuals_it). These residuals are the variation in a bond’s price informativeness

that are uncorrelated with the variation in that bond’s rebate. We then interact these residuals

with the fraction of insurer j’s cash collateral reinvestment portfolio that has a remaining

maturity of more than one year (Reinvestment riskjt). Intuitively, this new interaction term

tests whether price informativeness in the spot market affects the lending decision over and

above the variation already contained in the rebate. The inclusion of first-stage residuals as

a regressor means that the errors have a non-standard distribution. We bootstrapped

two-sided 99 percent confidence intervals by drawing the two-way insurer-bond cluster units with

replacement 9,999 times. The coefficient estimate on the interaction ofReinvestment risk_jt and

Rebateit) is virtually identical to the estimate obtained in Column 4. Using the bootstrapped

distribution, we can reject the null hypotheses that this coefficient is less or equal to zero at

less than 0.1 percent significance level. In addition, the coefficient on the new interaction term

is not statistically significant, indicating that there is no additional information from the spot

market beyond the effect of the rebate.

In conclusion, the results in Table 4 support our model’s prediction that securities lenders

with more aggressive cash collateral reinvestment strategies relax the limits of arbitrage. These

securities lenders’ willingness to lend to short sellers lowers the cost of taking short positions.

Intuitively, those lenders can afford to lend less profitable bonds because they make a higher

return by reinvesting the cash collateral they receive in more risky assets. A key implication of

this pattern of securities lending is that the pricing of those bonds will be more informative.

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5

Figures

6

Tables

Table 1: Summary statistics. Reinvestment risk is the fraction of the cash collateral reinvestment portfolio that has a remaining maturity of more than one year. Lending dummy is 1 when an insurer is lending a bond. Credit rating is the numerical translation and average of credit ratings across the three main agencies, setting AAA, or equivalent = 28, AA+ = 26, AA = 25, AA- = 24 ... CCC- = 9, CC = 7, and C = 4. Rebate is the amount paid by the securities lender to a securities borrower that provided cash collateral. The reinvestment risk index is the weighted average maturity transformation of all securities lenders holding a bond. Price informativeness is the negative of the bod-ask spread, the difference between the weighted average price a bond is sold by a dealer to client and the weighted average price the same bond is purchased by a dealer from a client. We take the negative of the spread to make the interpretation of the sign of the coefficients easier: The transformed variable is increasing in liquidity.

Obs Mean St. Dev. p25 Median p75

Bond-Year Data

Reinvestment risk index (insurers only) 31,875 0.18 0.16 0.06 0.14 0.25

Reinvestment risk index (all lenders) 31,875 0.13 0.17 0.02 0.07 0.17

Offering amount 30,627 1.27 3.66 0.4 0.6 1

Offering yield 24,856 75.83 11099.99 3.98 5.29 6.35

Remaining maturity (yrs) 30,422 10.74 9.66 5 7 12

Price informativeness 9,583 -0.64 0.81 -0.18 -0.44 -0.88

Log(daily volume) 12,517 -7.02 1.44 -8.1 -7.05 -5.99

% lendable held (Insurers shareit) 31,875 0.73 0.59 0.29 0.59 1

Amount outstanding 16,700 1.4 4.67 0.38 0.6 1

Credit rating 30,028 19.75 3.36 18 20 22

Weighted average rebate (Rebateit) 31,875 0.04 0.13 -0.01 0.03 0.08

HHI of life insurers’ holdings (HHIit) 31,875 0.17 0.27 0.02 0.07 0.2

Total lent/lendable (Market tightness_it) 31,875 0.05 0.11 0 0.01 0.05

Bond-Insurer-Year Data

Reinvestment risk 272,660 0.18 0.27 0 0 0.29

On loan dummy 335,710 0.07 0.25 0 0 0

Offering amount 327,183 1.12 2.62 0.4 0.65 1

Offering yield 269,504 31.45 6741.92 4.25 5.44 6.35

Remaining maturity 325,773 11.5 9.3 5 8 17

Amount outstanding 173,932 1.22 3.69 0.4 0.65 1.02

Credit rating 321,132 19.83 2.98 18 20 22

Weighted average rebate (Rebateit) 335,710 0.05 0.12 -0.01 0.03 0.08

% lendable held (Market shareijt) 335,710 0.07 0.11 0.01 0.03 0.08

HHI of life insurers’ holdings (HHIit) 335,710 0.17 0.26 0.03 0.07 0.19

Table 2: Correlation between bond price informativeness and life insurers’ cash collateral reinvestment. The unit of observation is a bond i at the end of year t. The dependent variable is the price informativeness of a bond (Price informativeness_it). The main explanatory variables are the reinvestment risk index (Reinvestment risk index_it) and the share of the bond held by insurers (Insurers shareit). Column 5 reports standard errors two-way

clustered by bond in parentheses. All other columns report robust standard errors. ***, **, and * represent statistical significance at the 1%, 5%, and 10% level, respectively.

Dependent variable:

Price informativeness_it (1) (2) (3) (4) (5)

Reinvestment risk indexit 0.285*** (0.064)

Reinvestment risk indexit 0.359*** 0.380*** 0.347** 0.347* (0.130) (0.128) (0.161) (0.191)

Insurers shareit -0.497*** -0.325*** -0.151** -0.151**

(0.032) (0.046) (0.059) (0.070)

HHIit -0.539*** -0.568*** -0.568***

(0.120) (0.146) (0.166)

Market tightness_it -0.122 -0.200 -0.200

(0.107) (0.124) (0.132)

Remaining maturity_it -0.031*** -0.031***

(0.002) (0.003)

Offering yield_i -0.065*** -0.065***

(0.007) (0.008)

Offering amount_i -0.058 -0.058

(0.046) (0.061)

Amount outstanding_it 0.207*** 0.207***

(0.046) (0.062)

Credit rating_it 0.014*** 0.014**

(0.005) (0.006)

Year FE Y Y Y Y Y

R2 0.027 0.074 0.078 0.224 0.224

Table 3: Correlation between bond trading volume and life insurers’ cash collateral reinvestment. The unit of observation is a bond i at the end of year t. The dependent variable is the log of daily trade volume (Ln(volume)_it). The main explanatory variables are the reinvestment risk index (Reinvestment risk index_it) and the share of the bond held by insurers (Insurers shareit). Column 5 reports standard errors two-way clustered by insurer and bond in

parentheses. All other columns report robust standard errors. ***, **, and * represent statistical significance at the 1%, 5%, and 10% level, respectively.

Dependent variable:

Ln(volume)_it (1) (2) (3) (4) (5)

Reinvestment risk indexit 0.960*** (0.091)

Reinvestment risk indexit 0.942*** 0.908*** 0.494*** 0.494*** (0.136) (0.136) (0.170) (0.183)

Insurers shareit -1.144*** -1.176*** -0.554*** -0.554***

(0.037) (0.055) (0.073) (0.084)

HHIit 0.108 0.053 0.053

(0.127) (0.157) (0.174)

Market tightness_it 2.657*** 1.864*** 1.864***

(0.172) (0.216) (0.240)

Remaining maturity_it 0.020*** 0.020***

(0.003) (0.003)

Offering yield_i -0.173*** -0.173***

(0.010) (0.012)

Offering amount_i -0.203*** -0.203**

(0.064) (0.099)

Amount outstanding_it 1.015*** 1.015***

(0.067) (0.094)

Credit rating_it -0.127*** -0.127***

(0.008) (0.010)

Year FE Y Y Y Y Y

R2 0.016 0.101 0.124 0.325 0.325

Table 4: Life insurers with more aggressive securities lending programs reduce the limits of arbitrage. The unit of observation is a bond i held by insurer j in year t. The dependent variable (Loanijt) is 1 when the bond is on loan and 0 otherwise. The main

explanatory variables are the fraction of the cash collateral reinvestment portfolio that has a remaining maturity of more than one year (Reinvestment riskjt) and the weighted average rebate

(Rebate_it). See Table 1 for a description of the other variables. Column 4 reports standard errors two-way clustered by insurer and bond in parentheses. The brackets in Column 5 contain the bootstrapped two-sided 99 percent confidence intervals calculated by drawing the two-way insurer-bond cluster units with replacement 9,999 times. Columns 1-3 report robust standard errors. ***, **, and * represent statistical significance at the 1%, 5%, and 10% level, respectively.

Dependent variable:

Loan

ijt

(1)

(2)

(3)

(4)

(5)

Reinvestment risk

jt

0.107***

0.108***

0.113***

0.113***

0.130

(0.009)

(0.012)

(0.009)

(0.031)

[0.08,0.18]

Reinvestment risk

jt

×

0.224***

0.265***

0.191***

0.191***

0.196

Rebate

it

(0.019)

(0.029)

(0.022)

(0.071)

[0.05,0.34]

Rebate

it

-0.376***

-0.298***

(0.011)

(0.022)

Reinvestment risk

jt

×

-0.013

PI residuals

it

[-0.03,0.0]

Market share

ijt

0.086***

(0.009)

HHI

it

-0.012***

(0.004)

Market tightness

_it

0.384***

(0.060)

Remaining maturity

_it

0.001***

(0.000)

Offering yield

_i

-0.002***

(0.000)

Offering amount

_i

-0.006*

(0.003)

Amount outstanding

_it

0.012***

(0.003)

Credit rating

_it

-0.002

(0.001)

Fixed effects:

Insurer, Year

Y

Y

Y

Y

Y

Bond issuer

N

Y

N

N

N

Bond

×

Year

N

N

Y

Y

Y

R

2

0.098

0.148

0.285

0.285

0.310